GBRidge: Generalized Bayesian Ridge Regression
In SafeBayes: Generalized and Safe-Bayesian Ridge and Lasso Regression
Description
Usage
Arguments
Details
Value
Author(s)
References
Examples
Description
The function GBRidge (Generalized Bayesian Ridge Regression) provides a Gibbs sampler to sample the posterior of generalized Bayesian ridge regression models with learning rate eta.
Usage
1
2
Arguments
y
Vector of outcome variables, numeric, NA allowed, length n.
X
Design matrix, numeric, dimension n x p, n >= 2.
eta
Learning rate eta, numeric, 0 <= eta <= 1. Default 1.
prior
List containing the following elements
prior$varE: prior for the variance parameter σ^2 with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chi-square distribution. Default (0,0).
prior$varBR: prior for the variance of the Gaussian prior for the coefficients beta, with parameters $df and $S for respectively degrees of freedom and scale parameters for an inverse-chi-square distribution. Default (0,0).
nIter
Number of iterations, integer. Default 1100.
burnIn
Number of iterations for burn-in, integer. Default 100.
thin
Number of iterations for thinning, integer. Default 10.
minAbsBeta
Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default 10^-9.
weights
Vector of weights, numeric, length n. Default NULL, in which case all weights are set to 1.
pIter
Print iterations, logical. Default TRUE.
Details
Details on generalized Bayesian regression can be found in (de Heide, 2016). The implementation is heavily based on the BLR package of (de los Campos et al., 2009).
Several authors have brought forward the idea of equipping Bayesian updating with a learning rate eta, resulting in an eta-generalized posterior (Vovk (1990), McAllester (2003), Seeger (2002), Catoni (2007), Audibert (2004), Zhang (2004)). Grunwald (2012) suggested its use as a method to deal with model misspecification. In the eta-generalized posterior, the likelihood is raised to the power eta in order to trade the relative weight of the likelihood and the prior, where eta = 1 corresponds to standard Bayes.
Value
$y
Vector of original outcome variables.
$weights
Vector of weights.
$mu
Posterior mean of the intercept.
$varE
Posterior mean of of the variance.
$yHat
Posterior mean of mu + X*beta + epsilon.
$SD.yHat
Corresponding standard deviation.
$whichNa
Vector with indices of missing values of y.
$fit$pD
Estimated number of effective parameters.
$fit$DIC
Deviance Information Criterion.
$bR
Posterior mean of beta.
$SD.bR
Corresponding standard deviation.
$prior
List containing the priors used.
$nIter
Number of iterations.
$burnIn
Number of iterations for burn-in.
$thin
Number of iterations for thinning.
$eta
Learning rate eta.
Author(s)
R. de Heide
References
de Heide, R. 2016.
The Safe-Bayesian Lasso. Master Thesis, Leiden University.
de los Campos G., H. Naya, D. Gianola, J. Crossa, A. Legarra, E. Manfredi, K. Weigel and J. Cotes. 2009.
Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: 375-385.
Audibert, J.Y. 2013.
Bayesian generalized double pareto shrinkage. Statistica Sinica 23 119-143.
Catoni, O. 2007.
PAC-Bayesian Supervised Classification. Lecture Notes - Monograph Series. IMS.
Grunwald, P.D. 2012.
chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, 2012. Proceedings. 169-183. Springer Berlin Heidelberg
McAllester, D. 2003.
PAC-Bayesian stochastic model selection. Machine Learning 51(1) 5-21
Vovk, V.G. 1990.
Aggregating strategies. In Proc. COLT 90 371-383
Zhang, T. 2004.
Learning bounds for a generalized family of Bayesian posterior distributions. Advances of Neural Information Processing Systems 16, Thrun, L.S. and Schoelkopf, B. eds., MIT Press 1149-1156
Examples
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rm(list=ls())
library(SafeBayes)
# Simulate data
x <- runif(100, -1, 1) # 100 random uniform x's between -1 and 1
y <- NULL
# for each x, a y that is 0 + Gaussian noise
for (i in 1:100) {
y[i] <- 0 + rnorm(1, mean=0, sd=1/4)
}
# Now sample 100 zero's and ones (coin toss)
cointoss <- sample(0:1, 100, replace=TRUE)
# indices of the ones
indices <- which(cointoss==1)
# we replace x and y with (0,0) for the indices the cointoss
# landed tail (1)
x[indices] <- 0
y[indices] <- 0
plot(x,y)
# Determine the generalized posterior for eta = 0.25
obj <- GBLasso(y, x, eta=0.25)
# posterior means of the coefficients beta and intercept mu
betafour <- obj$bL
mufour <- obj$mu
Example output
No prior was provided, improper priors are used.
Iter: 1 time/iter: 0.006 varE: 0.07 lambda: 17.519
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Iter: 2 time/iter: 0.001 varE: 0.033 lambda: 14.05
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Iter: 3 time/iter: 0 varE: 0.026 lambda: 2.819
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Iter: 4 time/iter: 0 varE: 0.031 lambda: 1.274
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Iter: 5 time/iter: 0 varE: 0.026 lambda: 1.24
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Iter: 6 time/iter: 0 varE: 0.023 lambda: 0.203
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Iter: 7 time/iter: 0 varE: 0.013 lambda: 1.298
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Iter: 8 time/iter: 0 varE: 0.031 lambda: 30.693
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Iter: 9 time/iter: 0 varE: 0.019 lambda: 22.173
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Iter: 10 time/iter: 0.001 varE: 0.039 lambda: 17.502
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Iter: 11 time/iter: 0 varE: 0.021 lambda: 28.285
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Iter: 12 time/iter: 0 varE: 0.031 lambda: 33.293
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Iter: 13 time/iter: 0 varE: 0.032 lambda: 11.306
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Iter: 14 time/iter: 0 varE: 0.022 lambda: 8.872
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Iter: 15 time/iter: 0 varE: 0.033 lambda: 4.443
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Iter: 16 time/iter: 0 varE: 0.019 lambda: 3.255
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Iter: 17 time/iter: 0 varE: 0.024 lambda: 1.33
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Iter: 18 time/iter: 0 varE: 0.023 lambda: 1.531
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Iter: 19 time/iter: 0 varE: 0.017 lambda: 1.377
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Iter: 20 time/iter: 0.001 varE: 0.015 lambda: 1.226
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Iter: 21 time/iter: 0 varE: 0.034 lambda: 0.526
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Iter: 22 time/iter: 0 varE: 0.028 lambda: 0.49
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Iter: 23 time/iter: 0 varE: 0.022 lambda: 0.696
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Iter: 24 time/iter: 0 varE: 0.021 lambda: 0.672
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Iter: 25 time/iter: 0 varE: 0.029 lambda: 1.025
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Iter: 26 time/iter: 0 varE: 0.02 lambda: 0.193
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Iter: 27 time/iter: 0.02 varE: 0.02 lambda: 3.385
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Iter: 28 time/iter: 0 varE: 0.042 lambda: 1.116
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Iter: 29 time/iter: 0 varE: 0.042 lambda: 1.818
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Iter: 30 time/iter: 0 varE: 0.015 lambda: 2.329
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Iter: 31 time/iter: 0 varE: 0.034 lambda: 4.574
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Iter: 32 time/iter: 0 varE: 0.026 lambda: 2.228
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Iter: 33 time/iter: 0 varE: 0.029 lambda: 3.719
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Iter: 34 time/iter: 0 varE: 0.023 lambda: 4.8
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Iter: 35 time/iter: 0 varE: 0.042 lambda: 28.956
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Iter: 36 time/iter: 0.001 varE: 0.024 lambda: 32.924
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Iter: 37 time/iter: 0 varE: 0.022 lambda: 30.338
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Iter: 38 time/iter: 0 varE: 0.027 lambda: 36.897
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Iter: 39 time/iter: 0 varE: 0.023 lambda: 59.602
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Iter: 40 time/iter: 0 varE: 0.05 lambda: 62.827
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Iter: 41 time/iter: 0 varE: 0.035 lambda: 25.14
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Iter: 42 time/iter: 0 varE: 0.035 lambda: 39.443
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Iter: 43 time/iter: 0 varE: 0.029 lambda: 29.187
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Iter: 44 time/iter: 0 varE: 0.034 lambda: 30.005
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Iter: 45 time/iter: 0 varE: 0.03 lambda: 38.683
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Iter: 46 time/iter: 0 varE: 0.033 lambda: 13.67
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Iter: 47 time/iter: 0.001 varE: 0.04 lambda: 26.218
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Iter: 48 time/iter: 0 varE: 0.024 lambda: 24.653
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Iter: 49 time/iter: 0 varE: 0.023 lambda: 23.225
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Iter: 50 time/iter: 0 varE: 0.021 lambda: 16.796
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Iter: 51 time/iter: 0 varE: 0.027 lambda: 10.008
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Iter: 52 time/iter: 0 varE: 0.026 lambda: 17.378
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Iter: 53 time/iter: 0 varE: 0.021 lambda: 14.535
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Iter: 54 time/iter: 0 varE: 0.03 lambda: 23.648
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Iter: 55 time/iter: 0 varE: 0.037 lambda: 47.287
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Iter: 56 time/iter: 0 varE: 0.048 lambda: 34.942
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Iter: 57 time/iter: 0 varE: 0.029 lambda: 72.146
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Iter: 58 time/iter: 0.001 varE: 0.041 lambda: 131.296
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Iter: 59 time/iter: 0 varE: 0.023 lambda: 99.184
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Iter: 60 time/iter: 0 varE: 0.044 lambda: 44.758
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Iter: 61 time/iter: 0 varE: 0.047 lambda: 85.964
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Iter: 62 time/iter: 0 varE: 0.032 lambda: 56.547
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Iter: 63 time/iter: 0 varE: 0.015 lambda: 57.707
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Iter: 64 time/iter: 0 varE: 0.026 lambda: 56.582
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Iter: 65 time/iter: 0 varE: 0.022 lambda: 8.147
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Iter: 66 time/iter: 0 varE: 0.033 lambda: 9.711
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Iter: 67 time/iter: 0 varE: 0.029 lambda: 4.224
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Iter: 68 time/iter: 0 varE: 0.071 lambda: 7.6
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Iter: 69 time/iter: 0.001 varE: 0.035 lambda: 9.819
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Iter: 70 time/iter: 0 varE: 0.027 lambda: 6.946
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Iter: 71 time/iter: 0 varE: 0.044 lambda: 3.858
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Iter: 72 time/iter: 0 varE: 0.022 lambda: 12.268
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Iter: 73 time/iter: 0 varE: 0.036 lambda: 5.337
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Iter: 74 time/iter: 0 varE: 0.045 lambda: 4.162
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Iter: 75 time/iter: 0 varE: 0.023 lambda: 3.26
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Iter: 76 time/iter: 0 varE: 0.048 lambda: 12.101
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Iter: 77 time/iter: 0 varE: 0.025 lambda: 16.817
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Iter: 78 time/iter: 0 varE: 0.028 lambda: 15.998
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Iter: 79 time/iter: 0.001 varE: 0.026 lambda: 11.76
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Iter: 80 time/iter: 0 varE: 0.028 lambda: 18.317
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Iter: 81 time/iter: 0 varE: 0.025 lambda: 25.527
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Iter: 82 time/iter: 0 varE: 0.027 lambda: 8.233
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Iter: 83 time/iter: 0 varE: 0.02 lambda: 16.851
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Iter: 84 time/iter: 0 varE: 0.026 lambda: 23.771
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Iter: 85 time/iter: 0 varE: 0.025 lambda: 7.179
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Iter: 86 time/iter: 0 varE: 0.041 lambda: 3.832
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Iter: 87 time/iter: 0 varE: 0.023 lambda: 4.967
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Iter: 88 time/iter: 0.001 varE: 0.031 lambda: 8.794
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Iter: 89 time/iter: 0 varE: 0.026 lambda: 29.104
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Iter: 90 time/iter: 0 varE: 0.018 lambda: 54.113
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Iter: 91 time/iter: 0 varE: 0.031 lambda: 31.029
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Iter: 92 time/iter: 0 varE: 0.028 lambda: 134.513
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Iter: 93 time/iter: 0 varE: 0.035 lambda: 166.556
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Iter: 94 time/iter: 0 varE: 0.021 lambda: 169.96
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Iter: 95 time/iter: 0 varE: 0.028 lambda: 1178.285
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Iter: 96 time/iter: 0.001 varE: 0.024 lambda: 1358.315
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Iter: 97 time/iter: 0 varE: 0.017 lambda: 240.717
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Iter: 98 time/iter: 0 varE: 0.032 lambda: 174.645
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Iter: 99 time/iter: 0 varE: 0.019 lambda: 313.672
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Iter: 100 time/iter: 0 varE: 0.023 lambda: 379.309
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Iter: 101 time/iter: 0 varE: 0.04 lambda: 857.419
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Iter: 102 time/iter: 0 varE: 0.027 lambda: 430.631
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Iter: 103 time/iter: 0.001 varE: 0.041 lambda: 245.527
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Iter: 104 time/iter: 0 varE: 0.04 lambda: 343.321
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Iter: 105 time/iter: 0 varE: 0.026 lambda: 161.739
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Iter: 106 time/iter: 0 varE: 0.02 lambda: 352.287
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Iter: 107 time/iter: 0 varE: 0.026 lambda: 115.5
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Iter: 108 time/iter: 0 varE: 0.044 lambda: 70.394
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Iter: 109 time/iter: 0 varE: 0.019 lambda: 110.681
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Iter: 110 time/iter: 0 varE: 0.033 lambda: 64.661
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Iter: 111 time/iter: 0.001 varE: 0.025 lambda: 61.562
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Iter: 112 time/iter: 0 varE: 0.037 lambda: 79.826
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Iter: 113 time/iter: 0 varE: 0.026 lambda: 23.152
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Iter: 114 time/iter: 0 varE: 0.033 lambda: 29.422
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Iter: 115 time/iter: 0 varE: 0.018 lambda: 33.439
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Iter: 116 time/iter: 0 varE: 0.023 lambda: 30.741
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Iter: 117 time/iter: 0 varE: 0.037 lambda: 38.964
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Iter: 118 time/iter: 0 varE: 0.028 lambda: 13.652
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Iter: 119 time/iter: 0.001 varE: 0.037 lambda: 37.315
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Iter: 120 time/iter: 0 varE: 0.024 lambda: 22.998
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Iter: 121 time/iter: 0 varE: 0.026 lambda: 21.199
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Iter: 122 time/iter: 0 varE: 0.031 lambda: 21.669
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Iter: 123 time/iter: 0 varE: 0.017 lambda: 13.739
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Iter: 124 time/iter: 0 varE: 0.036 lambda: 34.231
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Iter: 125 time/iter: 0 varE: 0.037 lambda: 32.852
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Iter: 126 time/iter: 0 varE: 0.028 lambda: 17.113
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Iter: 127 time/iter: 0.001 varE: 0.021 lambda: 7.031
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Iter: 128 time/iter: 0 varE: 0.026 lambda: 10.035
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Iter: 129 time/iter: 0 varE: 0.045 lambda: 10.437
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Iter: 130 time/iter: 0 varE: 0.033 lambda: 22.124
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Iter: 131 time/iter: 0 varE: 0.026 lambda: 5.235
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Iter: 132 time/iter: 0 varE: 0.026 lambda: 10.513
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Iter: 133 time/iter: 0 varE: 0.031 lambda: 10.895
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Iter: 134 time/iter: 0 varE: 0.035 lambda: 825.762
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Iter: 135 time/iter: 0.001 varE: 0.026 lambda: 401.53
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Iter: 136 time/iter: 0 varE: 0.016 lambda: 385.979
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Iter: 137 time/iter: 0 varE: 0.023 lambda: 639.687
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Iter: 138 time/iter: 0 varE: 0.026 lambda: 486.946
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Iter: 139 time/iter: 0 varE: 0.037 lambda: 989.997
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Iter: 140 time/iter: 0 varE: 0.033 lambda: 990.602
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Iter: 141 time/iter: 0 varE: 0.028 lambda: 1471.962
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Iter: 142 time/iter: 0 varE: 0.019 lambda: 452.838
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Iter: 143 time/iter: 0.001 varE: 0.024 lambda: 638.649
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Iter: 144 time/iter: 0 varE: 0.031 lambda: 767.355
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Iter: 145 time/iter: 0 varE: 0.036 lambda: 645.188
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Iter: 146 time/iter: 0 varE: 0.032 lambda: 1364.749
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Iter: 147 time/iter: 0 varE: 0.052 lambda: 1228.628
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Iter: 148 time/iter: 0 varE: 0.028 lambda: 393.565
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Iter: 149 time/iter: 0 varE: 0.039 lambda: 2368.706
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Iter: 150 time/iter: 0 varE: 0.025 lambda: 5294.669
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Iter: 151 time/iter: 0.001 varE: 0.021 lambda: 1085.275
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Iter: 152 time/iter: 0 varE: 0.019 lambda: 1097.409
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Iter: 153 time/iter: 0 varE: 0.037 lambda: 1447.079
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Iter: 154 time/iter: 0 varE: 0.052 lambda: 594.831
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Iter: 155 time/iter: 0 varE: 0.03 lambda: 905.968
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Iter: 156 time/iter: 0 varE: 0.019 lambda: 3039.567
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Iter: 157 time/iter: 0 varE: 0.018 lambda: 277.939
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Iter: 158 time/iter: 0 varE: 0.022 lambda: 52.862
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Iter: 159 time/iter: 0.001 varE: 0.024 lambda: 61.235
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Iter: 160 time/iter: 0 varE: 0.032 lambda: 18.111
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Iter: 161 time/iter: 0 varE: 0.048 lambda: 10.768
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Iter: 162 time/iter: 0 varE: 0.022 lambda: 14.253
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Iter: 163 time/iter: 0 varE: 0.032 lambda: 15.873
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Iter: 164 time/iter: 0 varE: 0.033 lambda: 16.438
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Iter: 165 time/iter: 0 varE: 0.028 lambda: 2.849
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Iter: 166 time/iter: 0.001 varE: 0.029 lambda: 1.431
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Iter: 167 time/iter: 0 varE: 0.016 lambda: 0.717
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Iter: 168 time/iter: 0 varE: 0.026 lambda: 3.144
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Iter: 169 time/iter: 0 varE: 0.028 lambda: 3.145
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Iter: 170 time/iter: 0 varE: 0.023 lambda: 4.287
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Iter: 171 time/iter: 0 varE: 0.019 lambda: 2.07
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Iter: 172 time/iter: 0 varE: 0.016 lambda: 1.535
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Iter: 173 time/iter: 0.001 varE: 0.022 lambda: 0.309
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Iter: 174 time/iter: 0 varE: 0.035 lambda: 0.235
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Iter: 175 time/iter: 0 varE: 0.027 lambda: 0.391
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Iter: 176 time/iter: 0 varE: 0.028 lambda: 0.553
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Iter: 177 time/iter: 0 varE: 0.045 lambda: 0.837
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Iter: 178 time/iter: 0 varE: 0.023 lambda: 0.687
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Iter: 179 time/iter: 0 varE: 0.035 lambda: 0.311
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Iter: 180 time/iter: 0 varE: 0.026 lambda: 0.657
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Iter: 181 time/iter: 0.001 varE: 0.03 lambda: 1.51
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Iter: 182 time/iter: 0 varE: 0.021 lambda: 0.949
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Iter: 183 time/iter: 0 varE: 0.029 lambda: 0.098
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Iter: 184 time/iter: 0 varE: 0.033 lambda: 0.118
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Iter: 185 time/iter: 0 varE: 0.026 lambda: 0.142
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Iter: 186 time/iter: 0 varE: 0.016 lambda: 0.87
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Iter: 187 time/iter: 0 varE: 0.036 lambda: 2.35
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Iter: 188 time/iter: 0 varE: 0.025 lambda: 2.433
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Iter: 189 time/iter: 0.001 varE: 0.021 lambda: 1.909
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Iter: 190 time/iter: 0.001 varE: 0.049 lambda: 1.154
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Iter: 191 time/iter: 0 varE: 0.023 lambda: 2.042
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Iter: 192 time/iter: 0 varE: 0.017 lambda: 1.012
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Iter: 193 time/iter: 0 varE: 0.016 lambda: 6.787
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Iter: 194 time/iter: 0 varE: 0.03 lambda: 8.204
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Iter: 195 time/iter: 0 varE: 0.024 lambda: 6.964
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Iter: 196 time/iter: 0 varE: 0.023 lambda: 0.757
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Iter: 197 time/iter: 0 varE: 0.038 lambda: 1.141
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Iter: 198 time/iter: 0 varE: 0.021 lambda: 2.371
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Iter: 199 time/iter: 0 varE: 0.021 lambda: 2.668
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Iter: 200 time/iter: 0.001 varE: 0.017 lambda: 2.445
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Iter: 201 time/iter: 0 varE: 0.018 lambda: 6.686
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Iter: 202 time/iter: 0 varE: 0.023 lambda: 13.857
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Iter: 203 time/iter: 0 varE: 0.05 lambda: 27.22
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Iter: 204 time/iter: 0 varE: 0.032 lambda: 13.057
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Iter: 205 time/iter: 0 varE: 0.017 lambda: 6.606
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Iter: 206 time/iter: 0 varE: 0.034 lambda: 10.29
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Iter: 207 time/iter: 0 varE: 0.024 lambda: 7.14
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Iter: 208 time/iter: 0 varE: 0.023 lambda: 21.92
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Iter: 209 time/iter: 0 varE: 0.013 lambda: 14.963
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Iter: 210 time/iter: 0 varE: 0.02 lambda: 3.023
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Iter: 211 time/iter: 0.001 varE: 0.025 lambda: 0.081
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Iter: 212 time/iter: 0 varE: 0.044 lambda: 0.236
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Iter: 213 time/iter: 0 varE: 0.025 lambda: 0.281
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Iter: 214 time/iter: 0 varE: 0.019 lambda: 0.326
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Iter: 215 time/iter: 0 varE: 0.032 lambda: 0.222
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Iter: 216 time/iter: 0 varE: 0.029 lambda: 0.603
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Iter: 217 time/iter: 0 varE: 0.027 lambda: 1.633
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Iter: 218 time/iter: 0 varE: 0.022 lambda: 0.697
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Iter: 219 time/iter: 0 varE: 0.017 lambda: 1.198
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Iter: 220 time/iter: 0 varE: 0.025 lambda: 0.554
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Iter: 221 time/iter: 0.001 varE: 0.032 lambda: 1.57
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Iter: 222 time/iter: 0 varE: 0.051 lambda: 1.107
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Iter: 223 time/iter: 0 varE: 0.047 lambda: 0.528
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Iter: 224 time/iter: 0 varE: 0.029 lambda: 1.913
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Iter: 225 time/iter: 0 varE: 0.051 lambda: 1.687
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Iter: 226 time/iter: 0 varE: 0.028 lambda: 12.898
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Iter: 227 time/iter: 0 varE: 0.018 lambda: 18.068
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Iter: 228 time/iter: 0 varE: 0.027 lambda: 14.48
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Iter: 229 time/iter: 0 varE: 0.029 lambda: 58.277
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Iter: 230 time/iter: 0 varE: 0.035 lambda: 40.688
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Iter: 231 time/iter: 0 varE: 0.027 lambda: 43.415
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Iter: 232 time/iter: 0.001 varE: 0.024 lambda: 76.447
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Iter: 233 time/iter: 0 varE: 0.026 lambda: 21.435
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Iter: 234 time/iter: 0 varE: 0.043 lambda: 18.063
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Iter: 235 time/iter: 0 varE: 0.017 lambda: 19.533
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Iter: 236 time/iter: 0 varE: 0.039 lambda: 39.811
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Iter: 237 time/iter: 0 varE: 0.037 lambda: 39.898
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Iter: 238 time/iter: 0 varE: 0.018 lambda: 28.236
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Iter: 239 time/iter: 0 varE: 0.03 lambda: 2.013
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Iter: 240 time/iter: 0 varE: 0.037 lambda: 1.156
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Iter: 241 time/iter: 0 varE: 0.028 lambda: 1.171
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Iter: 242 time/iter: 0.001 varE: 0.018 lambda: 1.305
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Iter: 243 time/iter: 0 varE: 0.046 lambda: 0.89
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Iter: 244 time/iter: 0 varE: 0.03 lambda: 3.098
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Iter: 245 time/iter: 0 varE: 0.025 lambda: 4.654
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Iter: 246 time/iter: 0 varE: 0.027 lambda: 2.471
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Iter: 247 time/iter: 0 varE: 0.03 lambda: 3.342
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Iter: 248 time/iter: 0 varE: 0.058 lambda: 2.516
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Iter: 249 time/iter: 0 varE: 0.016 lambda: 3.096
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Iter: 250 time/iter: 0 varE: 0.041 lambda: 5.729
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Iter: 251 time/iter: 0 varE: 0.033 lambda: 4.742
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Iter: 252 time/iter: 0.001 varE: 0.034 lambda: 5.483
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Iter: 253 time/iter: 0 varE: 0.024 lambda: 12.568
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Iter: 254 time/iter: 0 varE: 0.019 lambda: 0.305
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Iter: 255 time/iter: 0 varE: 0.033 lambda: 0.854
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Iter: 256 time/iter: 0 varE: 0.027 lambda: 0.698
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Iter: 257 time/iter: 0 varE: 0.034 lambda: 1.107
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Iter: 258 time/iter: 0 varE: 0.038 lambda: 1.088
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Iter: 259 time/iter: 0 varE: 0.021 lambda: 1.155
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Iter: 260 time/iter: 0 varE: 0.027 lambda: 1.362
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Iter: 261 time/iter: 0 varE: 0.025 lambda: 0.745
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Iter: 262 time/iter: 0.001 varE: 0.018 lambda: 4.958
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Iter: 1100 time/iter: 0.001 varE: 0.022 lambda: 6979719.918
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SafeBayes documentation built on May 1, 2019, 9:23 p.m.
R Package Documentation
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Description Usage Arguments Details Value Author(s) References Examples
Description
The function GBRidge (Generalized Bayesian Ridge Regression) provides a Gibbs sampler to sample the posterior of generalized Bayesian ridge regression models with learning rate eta.
Usage
1 2 |
Arguments
y |
Vector of outcome variables, numeric, NA allowed, length n. |
X |
Design matrix, numeric, dimension n x p, n >= 2. |
eta |
Learning rate eta, numeric, 0 <= eta <= 1. Default 1. |
prior |
List containing the following elements
|
nIter |
Number of iterations, integer. Default 1100. |
burnIn |
Number of iterations for burn-in, integer. Default 100. |
thin |
Number of iterations for thinning, integer. Default 10. |
minAbsBeta |
Minimum absolute value of sampled coefficients beta to avoid numerical problems, numeric. Default 10^-9. |
weights |
Vector of weights, numeric, length n. Default NULL, in which case all weights are set to 1. |
pIter |
Print iterations, logical. Default TRUE. |
Details
Details on generalized Bayesian regression can be found in (de Heide, 2016). The implementation is heavily based on the BLR package of (de los Campos et al., 2009).
Several authors have brought forward the idea of equipping Bayesian updating with a learning rate eta, resulting in an eta-generalized posterior (Vovk (1990), McAllester (2003), Seeger (2002), Catoni (2007), Audibert (2004), Zhang (2004)). Grunwald (2012) suggested its use as a method to deal with model misspecification. In the eta-generalized posterior, the likelihood is raised to the power eta in order to trade the relative weight of the likelihood and the prior, where eta = 1 corresponds to standard Bayes.
Value
$y |
Vector of original outcome variables. |
$weights |
Vector of weights. |
$mu |
Posterior mean of the intercept. |
$varE |
Posterior mean of of the variance. |
$yHat |
Posterior mean of mu + X*beta + epsilon. |
$SD.yHat |
Corresponding standard deviation. |
$whichNa |
Vector with indices of missing values of y. |
$fit$pD |
Estimated number of effective parameters. |
$fit$DIC |
Deviance Information Criterion. |
$bR |
Posterior mean of beta. |
$SD.bR |
Corresponding standard deviation. |
$prior |
List containing the priors used. |
$nIter |
Number of iterations. |
$burnIn |
Number of iterations for burn-in. |
$thin |
Number of iterations for thinning. |
$eta |
Learning rate eta. |
Author(s)
R. de Heide
References
de Heide, R. 2016. The Safe-Bayesian Lasso. Master Thesis, Leiden University.
de los Campos G., H. Naya, D. Gianola, J. Crossa, A. Legarra, E. Manfredi, K. Weigel and J. Cotes. 2009. Predicting Quantitative Traits with Regression Models for Dense Molecular Markers and Pedigree. Genetics 182: 375-385.
Audibert, J.Y. 2013. Bayesian generalized double pareto shrinkage. Statistica Sinica 23 119-143.
Catoni, O. 2007. PAC-Bayesian Supervised Classification. Lecture Notes - Monograph Series. IMS.
Grunwald, P.D. 2012. chapter The Safe Bayesian. Algorithmic Learning Theory: 23rd International Conference, ALT 2012, Lyon, France, October 29-31, 2012. Proceedings. 169-183. Springer Berlin Heidelberg
McAllester, D. 2003. PAC-Bayesian stochastic model selection. Machine Learning 51(1) 5-21
Vovk, V.G. 1990. Aggregating strategies. In Proc. COLT 90 371-383
Zhang, T. 2004. Learning bounds for a generalized family of Bayesian posterior distributions. Advances of Neural Information Processing Systems 16, Thrun, L.S. and Schoelkopf, B. eds., MIT Press 1149-1156
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | rm(list=ls())
library(SafeBayes)
# Simulate data
x <- runif(100, -1, 1) # 100 random uniform x's between -1 and 1
y <- NULL
# for each x, a y that is 0 + Gaussian noise
for (i in 1:100) {
y[i] <- 0 + rnorm(1, mean=0, sd=1/4)
}
# Now sample 100 zero's and ones (coin toss)
cointoss <- sample(0:1, 100, replace=TRUE)
# indices of the ones
indices <- which(cointoss==1)
# we replace x and y with (0,0) for the indices the cointoss
# landed tail (1)
x[indices] <- 0
y[indices] <- 0
plot(x,y)
# Determine the generalized posterior for eta = 0.25
obj <- GBLasso(y, x, eta=0.25)
# posterior means of the coefficients beta and intercept mu
betafour <- obj$bL
mufour <- obj$mu
|
Example output
No prior was provided, improper priors are used.
Iter: 1 time/iter: 0.006 varE: 0.07 lambda: 17.519
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Iter: 2 time/iter: 0.001 varE: 0.033 lambda: 14.05
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Iter: 1099 time/iter: 0 varE: 0.032 lambda: 17570796.613
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Iter: 1100 time/iter: 0.001 varE: 0.022 lambda: 6979719.918
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